Bending of a Light-Ray Passing a Black Hole
| Scientific Paper | |
|---|---|
| Title | Bending of a Light-Ray Passing a Black Hole |
| Read in full | Link to paper |
| Author(s) | Dan Romalo |
| Keywords | bending of light-rays, black hole |
| Published | 2005 |
| Journal | Proceedings of the NPA |
| Volume | 2 |
| No. of pages | 12 |
| Pages | 158-164 |
Read the full paper here
Abstract
Assuming the hypothesis of a 1/r2 law of ether absorption-speed in the vicinity of massive celestial bodies and using an elementary, intuitive, calculus procedure, the bending of a light-ray passing near a black hole is evaluated. The theoretical results so obtained are, in some aspects, stunning because they suggest that some strong astronomic anomalies may, or should be observable by adequate means. And also because some accepted cosmologic fundamentals may become questionable. The so arisen problems are just suggested without any pretense to be rigid statements.
This paper is aka "Most Elementarily-Intuitive, Tentative Approach to Evaluate the Bending of a Light-Ray Passing a Black Hole"
Overview
Dan Romalo attempts to reproduce gravitational light deflection without general relativity, using instead a flowing material ether. The hypothesis, which he credits to Herbert E Ives "in his last years" and develops further in a companion paper for Galilean Electrodynamics, is that a universal ether is continuously absorbed by massive bodies, so that space is filled with an inflow field "the speed distribution being the classic one of a sink in a non compressible fluid". Light then propagates classically within that moving medium: the wave-front is carried bodily by the flow, and because the flow speed differs across the width of a beam the wave-front also rotates. The trajectory is obtained by stepping the ray forward in small increments and accumulating both effects.
The calculation is deliberately elementary — the whole model is an Excel spreadsheet, and the paper reproduces its columns formula by formula. Romalo is candid that this is exploratory: the conclusions "are just suggested without any pretense to be rigid statements", and he closes by surrendering the ground to astronomers. What he finds, however, he regards as startling: light approaching a black hole is first bent away from it before turning back; beams passing at a few hundred metres may loop more than once around the hole and be trapped; and every black hole should therefore generate a virtual duplicate image of every source in the sky, so that the universe may contain far fewer objects, and far less mass, than astronomers count.
The argument
The model
Two physical assumptions are stated at the outset: that light propagation at the macroscopic level is "100% of undulatory nature", and that light travels perpendicular to its wave-front element, which is itself carried by an ether flow drained into a sink at the black hole's centre. Nothing else is assumed — no curved space-time, no metric, no photon.
The geometry is polar, centred on the hole, with θ the angular position of the ray, r its radius, y0 the initial offset from the axis, h the beam width (taken as 8 m, "of the order of a big telescope's aperture") and ε the running angle between the ray's direction and the local radius. Stepping in equal increments Δθ = 0.001 rad, elementary trigonometry gives the two relations Romalo uses to advance the trajectory,
ri = ri+1cos Δθ + Δs cos(θi + ε) and ri+1 sin Δθ = Δs sin(θi + ε),
from which, eliminating Δs, ri+1 = ri sin(θi + ε)/sin(θi+1 + ε), and Δsi = riΔθ / sin(θi + Δθ + ε).
Advection and wave-front rotation
The step length is converted to a time interval by combining the light speed c relative to the ether with the ether transport velocity, resolved into components: (Δs)x = (c cos εi + ½(vr + vr+h) cos θi)Δt and (Δs)y = (c sin εi − ½(vr + vr+h) sin θi)Δt, with Δt = Δs/va and va the resultant. The inflow speeds at the two edges of the beam are computed, in the spreadsheet columns as printed, as vr = √(2kM/r) and vr+h = √(2kM/(r+h)), with the radius at the far edge given by r +̃ h = r + h sin(θi + εi).
The bending itself comes from the difference between those two speeds. To first order, tan Δεi = (vr − vr+h)i cos θi Δt/h, so one edge of the wave-front is dragged faster than the other and the beam pivots. The beam width h cancels out of the accumulated result, as it must. Summing arctan Δε over the steps gives the running deflection εi, and yi = ri sin θi traces the shape of the path. Romalo notes that his scheme determines ε one step late, and argues this does not break its coherence.
Results
Table 1 records runs at fixed M = 1031 kg — "5 times the sun's mass" — with ε0 = 0 and y0 varied from 0.1 m to 1011 m. For the smallest offsets a preliminary phase of 10,000 steps at 10−7 rad per step is prepended, since r0 would otherwise start "inconclusively small".
Three findings are highlighted:
- Initial repulsion. "On approaching the black hole, the beam is always bent in the away direction", passing at yπ/2 > y0 with επ/2 > 0. Only afterwards does a "bend-towards" phase set in, driving ε negative and settling the ray onto a final asymptote. The observer therefore sees the source not in its true position but at a virtual one.
- Ring and double images. With source, hole and observer nearly aligned, beams passing at 1.7×107 to 108 m should build "a virtual image shaped as a fuzzy ring" of angular diameter between 4.656×10−4 and 1.338×10−6 rad; slight misalignment yields two images straddling the hole, at apparent linear separation d = ras sin αas with αas = θas − π − εas. Romalo asks whether this could explain a Hubble image reproduced from National Geographic.
- Capture. Near y0 = 700 m the bending becomes "quite 'monstrous'", with θas exceeding 2π: the ray loops more than once around the hole, and may end on a "closed, most probably wavy, nearly circular trajectory".
The cosmological consequence
From the second finding Romalo draws his "terrible" conclusion. If every black hole yields at least one extra virtual image of every luminous object, the sky is populated by as many as (2n − 1)N spurious sources for n holes and N real objects. Even allowing that most would be too faint, too red-shifted or too delayed to see, "the number of cosmic objects actually seen would... still largely exceed the number of the really existing ones", and hence "the total mass contained into the known universe could be much less than the till now accepted as being the real one". He suggests that the basis for evaluating the cosmological constant is thereby in question, and that assuming the constancy of both c and G is "definitely controversial" inside an ether framework.
A closing speculation asks whether radiation trapped on circular orbits, accumulating without compensating emission, must organise into a coherent mode and re-radiate, so that "every black hole in the sky would 'glow' with an individual-specific, quasi-monochromatic radiation" in the radio band — checkable, he suggests, against existing radio-telescope archives. He appends an acknowledgement to Dr Cynthia Kolb Whitney "for observing and warning me that my hypothesis on 'spiral-capturing of electromagnetic waves by black holes' does not respect a fundamental principle of optics."
Assessment
There is something genuinely appealing in the attempt. A sink flow in an ether is a concrete, visualisable mechanism, and the idea that light bends because a velocity gradient across the wave-front rotates it is sound physics in its own domain — it is how refraction works in an inhomogeneous medium, and the beam width h correctly cancels from the answer. Romalo also does not oversell: he flags the one-step lag in his ε recursion, admits the ring result is "without these data being a really conclusive proof", and prints his acknowledgement that a referee found a fundamental optical objection to the capture speculation — almost certainly the conservation of étendue, which forbids a passive system from concentrating radiation to a brightness above the source's. Publishing that objection alongside the speculation is creditable.
The difficulties are, unfortunately, arithmetic as much as conceptual, and the first one is internal. The abstract announces "a 1/r2 law of ether absorption-speed", consistent with flux conservation into a sink in an incompressible fluid. But the spreadsheet computes vr = √(2kM/r), which is the Newtonian escape velocity and varies as r−1/2. These are not the same hypothesis, and nothing in the paper reconciles them. Whichever was intended, the published numbers were produced by the second.
The scale of the black hole can be checked directly. For M = 1031 kg the Schwarzschild radius is 2GM/c2 ≈ 1.49×104 m, the photon sphere sits at 2.2×104 m, and the critical impact parameter for capture is (3√3/2)(2GM/c2) ≈ 3.9×104 m. Romalo's model puts the onset of looping and trapping near y0 = 700 m — some fifty times too close in. (His "5 times the sun's mass" is right: 1031 kg is 5.03 M☉.)
The deflections themselves fare worse. Both Newtonian and relativistic gravitational bending fall off as 1/b; the relativistic value is 4GM/c2b = 2.97×104 m / b for this mass. Reading Table 1, εas = −2.629×10−3 rad at y0 = 106 m, where the relativistic value is 2.97×10−2 rad — about eleven times larger. At y0 = 1011 m the table gives −6.69×10−7 rad against a relativistic 2.97×10−7 — now the model is more than twice as large. The tabulated values do not follow a 1/b law at all: between those two rows y0 grows by 105 while εas falls by only 3.9×103. A model of gravitational deflection whose own output is not asymptotically inverse in the impact parameter has a numerical problem, and the most likely culprit is accumulation error in a fixed Δθ = 0.001 rad integration run over thousands of steps in single-precision spreadsheet arithmetic. Until that is resolved the "monstrous" bending at 700 m cannot be distinguished from a divergence of the integrator.
The initial "bend-away" phase is the most striking claim and the one most directly at odds with measurement. No repulsive phase appears in solar light deflection, which has been measured by very-long-baseline interferometry and by the Cassini Doppler tracking experiment to agree with the relativistic prediction to about two parts in 104; a model that departs from that prediction by factors of two to eleven, in either direction, is excluded by those data long before one reaches a black hole. Romalo does not compare his numbers with the solar case at all, which is the obvious available test and one his own spreadsheet could have run.
Finally, the cosmological inference does not follow even granting the optics. Multiple imaging by a compact deflector requires close alignment of source, lens and observer — a condition Romalo himself states in case (a) — and the resulting cross-section is minute, which is why observed strong lenses are rare rather than universal. Counting one duplicate image per black hole per source, as the (2n − 1)N estimate does, assigns every hole an alignment it almost never has. Gravitational lensing is real and does produce Einstein rings and multiple quasar images much as the paper's case (a) describes; but it is measured, modelled, and used to weigh mass rather than to subtract it, and the observed image multiplicities are far too rare to inflate the source counts on which cosmological mass estimates rest. Read as what its author calls a "brain-storming exercise", the paper is an honest and readable attempt; read as a calculation, its own table does not yet support its conclusions.